Theorem grpsubfvalg 13800

Description: Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 19-Feb-2024.)

Hypotheses

Ref	Expression
grpsubval.b	|- B = ( Base ` G )
grpsubval.p	|- .+ = ( +g ` G )
grpsubval.i	|- I = ( invg ` G )
grpsubval.m	|- .- = ( -g ` G )

Assertion

Ref	Expression
grpsubfvalg	|- ( G e. V -> .- = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )

Distinct variable groups:   x, y, B   x, G, y   x, I, y   x, .+ , y

Allowed substitution hints:   .- ( x, y)   V( x, y)

Proof of Theorem grpsubfvalg

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpsubval.m	. 2 |- .- = ( -g ` G )
2		df-sbg 13760	. . 3 |- -g = ( g e. _V |-> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( +g ` g ) ( ( invg ` g ) ` y ) ) ) )
3		fveq2 5675	. . . . 5 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
4		grpsubval.b	. . . . 5 |- B = ( Base ` G )
5	3, 4	eqtr4di 2285	. . . 4 |- ( g = G -> ( Base ` g ) = B )
6		fveq2 5675	. . . . . 6 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
7		grpsubval.p	. . . . . 6 |- .+ = ( +g ` G )
8	6, 7	eqtr4di 2285	. . . . 5 |- ( g = G -> ( +g ` g ) = .+ )
9		eqidd 2235	. . . . 5 |- ( g = G -> x = x )
10		fveq2 5675	. . . . . . 7 |- ( g = G -> ( invg ` g ) = ( invg ` G ) )
11		grpsubval.i	. . . . . . 7 |- I = ( invg ` G )
12	10, 11	eqtr4di 2285	. . . . . 6 |- ( g = G -> ( invg ` g ) = I )
13	12	fveq1d 5677	. . . . 5 |- ( g = G -> ( ( invg ` g ) ` y ) = ( I ` y ) )
14	8, 9, 13	oveq123d 6079	. . . 4 |- ( g = G -> ( x ( +g ` g ) ( ( invg ` g ) ` y ) ) = ( x .+ ( I ` y ) ) )
15	5, 5, 14	mpoeq123dv 6123	. . 3 |- ( g = G -> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( +g ` g ) ( ( invg ` g ) ` y ) ) ) = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )
16		elex 2827	. . 3 |- ( G e. V -> G e. _V )
17		basfn 13355	. . . . . 6 |- Base Fn _V
18		funfvex 5692	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
19	18	funfni 5463	. . . . . 6 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
20	17, 16, 19	sylancr 414	. . . . 5 |- ( G e. V -> ( Base ` G ) e. _V )
21	4, 20	eqeltrid 2321	. . . 4 |- ( G e. V -> B e. _V )
22		mpoexga 6421	. . . 4 |- ( ( B e. _V /\ B e. _V ) -> ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) e. _V )
23	21, 21, 22	syl2anc 411	. . 3 |- ( G e. V -> ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) e. _V )
24	2, 15, 16, 23	fvmptd3 5776	. 2 |- ( G e. V -> ( -g ` G ) = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )
25	1, 24	eqtrid 2279	1 |- ( G e. V -> .- = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   _Vcvv 2815   Fn wfn 5352   ` cfv 5357  (class class class)co 6058   e. cmpo 6060   Basecbs 13296   +g cplusg 13374   invgcminusg 13756   -gcsg 13757

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-inn 9255  df-ndx 13299  df-slot 13300  df-base 13302  df-sbg 13760

This theorem is referenced by:  grpsubval  13801  grpsubf  13834  grpsubpropdg  13859  grpsubpropd2  13860